When I think of the Red River, I think of the border between Texas and Oklahoma, but you're right. The Red River also forms a small part of the border between Texas and Arkansas, too. Texarkana, TX, just south of the river, actually straddles the border of the two states.
In fact, a Texas wildlife agent was patrolling the border when he came across this coyote caught in a bear trap. So he got on his radio and called up his counterpart from the Arkansas side. Said, "Hey, I got one of your coyotes over here caught in a bear trap. You'd better come over and get him."
The Arkansas game ranger answered, "He's on your side of the border; you deal with it."
"Nope," answered the Texan, "it's your coyote so it's your responsibility."
From the Arkansas side, "How are you so sure it's one of OUR coyotes?"
Answered the Texan, "Well, he's already chewed off three of his legs trying to get free and he's STILL stuck in the trap...!"